We present a method that employs physics-informed deep learning techniques for parametrically solving partial differential equations. The focus is on the steady-state heat equations within heterogeneous solids exhibiting significant phase contrast. Similar equations manifest in diverse applications like chemical diffusion, electrostatics, and Darcy flow. The neural network aims to establish the link between the complex thermal conductivity profiles and temperature distributions, as well as heat flux components within the microstructure, under fixed boundary conditions. A distinctive aspect is our independence from classical solvers like finite element methods for data. A noteworthy contribution lies in our novel approach to defining the loss function, based on the discretized weak form of the governing equation. This not only reduces the required order of derivatives but also eliminates the need for automatic differentiation in the construction of loss terms, accepting potential numerical errors from the chosen discretization method. As a result, the loss function in this work is an algebraic equation that significantly enhances training efficiency. We benchmark our methodology against the standard finite element method, demonstrating accurate yet faster predictions using the trained neural network for temperature and flux profiles. We also show higher accuracy by using the proposed method compared to purely data-driven approaches for unforeseen scenarios.

翻译：本文提出一种基于物理信息深度学习技术的偏微分方程参数化求解方法，重点研究具有显著相衬度的非均质固体中的稳态热传导方程。此类方程同样适用于化学扩散、静电学和达西流等领域的多种应用场景。神经网络旨在建立固定边界条件下复杂热导率分布与微观结构中温度场及热通量分量之间的映射关系。本方法的一个显著特征是无需依赖有限元法等传统数值求解器生成训练数据。关键创新在于基于控制方程的离散弱形式定义损失函数：这不仅降低了所需导数的阶数，更消除了在构建损失项时对自动微分的依赖——尽管引入了所选离散化方法可能带来的数值误差。由此，本文提出的损失函数实质为代数方程，显著提升了训练效率。我们将所提方法与标准有限元法进行基准对比，证明训练后的神经网络能够以更高效率实现温度场与热通量分布的精确预测。此外，面对未知工况时，本方法相较于纯数据驱动方法展现出更高的预测精度。